8. (sta) b s²+6² 10. e-sD 11. 9. s+a (s+a)²+6² where a and n are constants. where b is a constant. where D is a constant. where a and b are constants.

Q: 3. Let d 2 2 and let P(z) = ao + ajz + ... + a4z“, ad # 0 be a polynomial with real coefficients…

A: Please note that I am able to see only one part, namely part (a). So, I can solve only part (a) for…

Q: (b) Using the Residue Theorem, or otherwise, compute z-7 -z-2 dz, where C is the positively oriented…

A: Useful result:

Q: Theorem (Fundamental Theorem of Algebra). A polynomial p(z) = anz" +.+a¡z+ ao with complex…

A: Note that any polynomial with real coefficient is always continuous. We will use this in step 2 to…

Q: Given the two points (x, V) and (x, y), the Lagrange polynomial is P (x) that passes through X-X.…

A:

Q: nxn For any A, B in R prove that Įrank(A) - rank(B)| ≤ rank(A + B) ≤ rank (A) + rank(B). }

A:

Q: 18. Find a monic polynomial of least degree in R[x] that has 4i-1 and -3 as roots. write the powers…

A:

Q: 49) Let P(z) = ao + a1z +……. 1 order n at o∞. Consider + anz" be a polynomial of degree n. Show that…

A: Given that Pz=a0+a1z+a2z2+⋯+anzn be a polynomial of degree n.Then, an≠0. We have to show that 1P(z)…

Q: can you explain how you got (x-2)=(t-3) and can you explain how you got the new limits because the…

A: Given:

Q: When using yi=f(xi) for i=0,1, ..., n to find the Lagrange interpolating polynomial ---p(x), p(xi)…

A:

Q: lim (z,y)→(0,0) (3 + (x² + y²) In(x² + y?)) 00 does not exist O 30 1,

A:

Q: x = 10 (s+2) (s³+ s)'

A:

Q: (b) Using the Residue Theorem, or otherwise, compute Sezz z-7 -22²-22-2 dz, where C is the…

A: We shall first split the integral into partial fractions and then use Cauchys integral formula

Q: Evaluate by the residue theorem: foodz, 0<x< 1 1+x sin(an)

A:

Q: Consider the functi on f(z) = 2 -a with lal >1 %3D 1- az Show that if 121 >1, the lf(Z)< 1

A: Given the function f(z)=z-a1-a¯z with |a|&gt;1.To prove that if |z|&gt;1, then |f(z)|&lt;1

Q: Discu ss the Singu laritie s and poles of zez Sin z f (z) = (z-2)(z²-1)* (2²+52+6) (z+7) (z²+4)

A:

Q: (a) Let xo, ₁,..., n be n+1 distinct points in [a, b], with co= a and cn = b, and f € C+¹[a, b]. Let…

A:

Q: Evaluate the improper intergal 1 dr using the residue theorem. (x² + 4)?

A: Solving imroper integral using residue theorem

Q: Using residue theorem to evaluate f (z+1)2 zp

A:

Q: Use the Residue Theorem to find the following (a) ez dz 22 – 1 J]z-1|=1 (b) e dz Izl=42 22 – 1

A: *concept:- Residue theorem.

Q: Let f(x) = x' + 2x + 4and g(x)= 3x + 2 in Z,[»). Then, upon dividing f(x) by g(x), we get 7- The…

A:

Q: b

A: Part (a): exp(-z) / (z^2 - 1)Identify the singularities: The function has simple poles at z = 1 and…

Q: ;0<x<0, - 0 <0 <0 Let X,,X,X,r(x;e)=- e.w Discuss the sufficiency by using Factorization Theorem.

A:

Q: As mentioned in class on Tucsday, the Legendre polynomials are a set of polynomial functions defined…

A: A function f(x) is said to be odd if it satisfies the condition f(-x)=-f(x), similarly the function…

Q: 6. Let f(r) be a function such that: |S(CD|SE-D | SED -1) | f'(-1) | ƒ"(-1) | f"(-1) | ƒ(¹) (-1)…

A:

Q: Find a monic polynomial of least degree in R[x] that has 1-i and 2i as roots.

A:

Q: By using Ho (X) = 1 = 1 and HI (X)=2x in Hermit polynomials, calculate the value of Hg (1) ?

A: The Hermite polynomial is a particular set of orthogonal polynomials. According to the definition…

Q: a) what (seriz = cosz ; are the residues of IT csc(Tz) Cs 650 3 = गर्नेछ ) sin

A: Note- Since you have posted multiple question we will solve only first part as per guideline repost…

Q: At least one of the answers above is NOT correct. Let f(z) = 2z¹ +22³ + 37² +2z + 1 and g(z) = 21³ +…

A:

Q: 20. Find the splitting field for x^2-2sqrt(2)x+3 over Q(sqrt(2)). Find the roots first then…

A: To Find: Splitting field of x2-22x+3=0 over ℚ2.

Q: Factor f(x) = x¹³ - x € Q[x] completely into irreducible factors. Hint: Factor out x first. The…

A:

Q: 11. The Hermite polynomials, H,„(x), satisfy the following: į. = Se-**H„(x)Hm(x) dx = \T2"n! 8,m:…

A: To prove: ∫-∞∞xe-x2HnxHmxdx=π2n-1n![δm,n-1+2n+1δm,n+1] Proof: Take the LHS ∫-∞∞xe-x2HnxHmxdx and…

Q: X (5) Give an example of an abstract simplicial complex such that Geo (x) 2 is & 1-simplex in R²…

A: a) One example of an abstract simplicial complex X such that Geo(X) is a 1-simplex in R2 is the…

Q: Find all the real zeros of the polynomial P(r) r'+6r+7x-4.

A: Given,          The polynomial is Px=x3+6x2+7x-4 .

Q: Prove that Using Cauchy's Residue Theorem 0 sin² (0) a + bcos (0) -d0 = 2π 62 -[a- √a² - 6²]

A: